[hamdani yusuf (OP)]

While reading history of scientific progress I got interested to the discovery of planet  Neptune as confirmation for the validity of Newton's gravity. This led Urban LeVerrier to hypothesize the existence of planet Vulcan to explain precession of Mercury’s perihelion.
I'm curious at how LeVerrier got his results, and after some searching I found this article.

http://www.mathpages.com/home/kmath280/kmath280.htm

But then I realized that the methods described there involve some approximations, even though we are looking for very small discrepancy (43 arcseconds per century), perhaps due to computational limitation of that time.



Does anyone know about more accurate method to calculate Newtonian precession of Mercury’s perihelion?
What is the actual difference compared to observation?

[evan_au]

Rather than approximating Jupiter's orbit with a ring having the mass of Jupiter, you could actually simulate the solar system numerically, comparing the motion of the planets using Newton's gravity, with or without Einstein's tweak for high-gravity environments.

Apparently, Newton originally assumed that gravity had a small inverse-cube component, which is not believed to exist. The oblate shape of the Sun does produce a very small inverse-cube component.

In the early days, there was some uncertainty about the distribution of mass within the Sun, but I expect that this has now been resolved with the aid of Helioseismology.

See: https://en.wikipedia.org/wiki/Apsidal_precession

[hamdani yusuf (OP)]

How did the discrepancy between observation and Newtonian calculation lead to hypothesis of planet Vulcan?
Where is the location it supposed to be, and what's the mass to compensate the calculation discrepancy?

[evan_au]

How did the discrepancy between observation and Newtonian calculation lead to hypothesis of planet Vulcan?

A discrepancy between the orbit of Uranus and predictions from Newton's theory of gravitation led to the mathematical prediction and discovery of a new planet, Neptune, in the 1840s.

The same theory is now guiding the current search for a hypothetical Planet Nine, to replace the demoted Pluto.

Where is the location (Vulcan) supposed to be?

The original idea (in 1842) was that Vulcan would orbit between Mercury and the Sun.
But a test of the theory (in 1843) did not line up with reality, so it was already in doubt.
However, the theory was kept alive by various astronomers who claimed to have seen the non-existent planet.

See: https://en.wikipedia.org/wiki/Vulcan_(hypothetical_planet)